Furthermore, sometimes it may be necessary to find a unit vector given some other vector as reference. If we have some vector $\vec{u}$, we can find a unit vector $\vec{u}_{unit}$ that goes in the same direction as $\vec{u}$ with the following formula:

For example, consider the vector $\vec{u} = (3, 4)$. We note that $\| \vec{u} \| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$. If we wanted to find a unit vector that went in the same direction as $\vec{u}$, all we would do is apply our formula, that is $\vec{u}_{unit} = \vec{u}\frac{1}{\| \vec{u} \|} = (3, 4) \cdot \frac{1}{5} = (\frac{3}{5}, \frac{4}{5})$.

Example 1

Find a unit vector that goes in the same direction as vector $\vec{u} = (1, 2, 3)$ and then verify this new vector has a magnitude of $1$ and goes in the same direction as $\vec{u}$.

Now we know that if $k$ is a scalar, then the vector $k\vec{u}$ will go in the same direction as $\vec{u}$. In this case, our scalar is $\frac{1}{\| \vec{u} \|}$, so our result will go in the same direction as our original vector.

Example 2

Given the vector $\vec{a} = (a_1, \frac{1}{2}, \frac{1}{3})$, give all values for the component $a_1$ such that $\| \vec{a} \| = 1$.

Since $\vec{u} \in \mathbb{R}^3$, we substitute into the formula for the norm of a vector to obtain: